login
Bisection of Chebyshev {S(n, 5)}_{n>=0}; the even part.
0

%I #10 May 28 2023 11:39:06

%S 1,24,551,12649,290376,6665999,153027601,3512968824,80645255351,

%T 1851327904249,42499896542376,975646292570399,22397364832576801,

%U 514163744856696024,11803368766871431751,270963317893186234249

%N Bisection of Chebyshev {S(n, 5)}_{n>=0}; the even part.

%C The odd part of this bisection is given by 5*A097778(n), for n >= 0.

%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>`

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (23,-1).

%F a(n) = S(2*n, 5) = S(n, 23) + S(n-1, 23), with the Chebyshev S polynomials (see A049310), S(-1, x) = 0, S(n, 5) = A004254(n+1) and S(n, 23) = A097778(n).

%F O.g.f.: (1 + x)/(1 - 23*x + x^2).

%F a(n) = 23*a(n-1) - a(n-2), for n >= 0, with a(-1) = -1 and a(-2) = -24.

%t Table[ChebyshevU[2*n, 5/2], {n, 0, 20}] (* _Vaclav Kotesovec_, May 27 2023 *)

%o (PARI) a(n) = polchebyshev(2*n, 2, 5/2); \\ _Michel Marcus_, May 27 2023

%Y Cf. A004254, A049310, A097778.

%K nonn,easy

%O 0,2

%A _Wolfdieter Lang_, Apr 26 2023